One Innovative Method for Improving the Power Density and Efficiency of Electro-Hydrostatic Actuators

Abstract

Although electro-hydrostatic actuators (EHAs) hold broad application prospects in more-electric aircraft and high-end equipment, they face a difficult trade-off between dynamic response and energy efficiency. To simultaneously enhance the dynamic response and energy efficiency of the EHA, this paper designs an innovative variable pump displacement and variable motor speed (VPVM) configuration that utilizes an electro-hydraulic servo valve for active displacement control. To address the flow mismatch problem associated with traditional asymmetric single-rod cylinders without reducing the power density of EHA, this paper also designs an innovative symmetric single-rod cylinder configuration. Based on the above two innovative configurations, this paper further develops a corresponding EHA prototype with a rated power density of 0.72 kW/kg. Simulation and experimental results demonstrate that compared to the traditional EHA with the fixed pump displacement and variable motor speed configuration (FPVM-EHA), the EHA with the proposed VPVM configuration (VPVM-EHA) not only improves energy efficiency and reduces motor heat generation under low-speed and heavy-load conditions, but also achieves a dynamic response close to that of the FPVM-EHA under fast dynamic response conditions.

1. Introduction

Electro-hydrostatic actuators (EHAs) are partially enclosed hydraulic systems that integrate components such as the motor, pump, tank, valve block, and cylinder [1,2], with the advantages of high power density, high efficiency, and easy maintenance. EHAs have broad application prospects in high-end equipment [3,4,5]. According to the different control variables, EHA configurations can be divided into three types: fixed pump displacement and variable motor speed (FPVM) [6], variable pump displacement and fixed motor speed (VPFM) [7], and variable pump displacement and variable motor speed (VPVM) [8]. Due to the advantages of simple structure and high reliability of FPVM configuration, most EHAs currently adopt this configuration [9,10]. However, EHAs with FPVM configurations (referred to as FPVM-EHA) are prone to motor overheating under low-speed and heavy-load conditions, which not only reduces energy efficiency but also shortens service life [11]. Some studies have also reported that under certain operating conditions, FPVM-EHAs generate less heat than EHAs with VPFM configuration (VPFM-EHAs) [12], which highlights the importance of selecting the configuration according to the application scenario.

One of the current challenges in the development of EHAs is the trade-off between high dynamic response and high energy efficiency, that is, improving dynamic response often comes at the expense of energy efficiency, and vice versa [13,14]. To simultaneously improve dynamic response and energy efficiency, researchers have proposed the VPVM configuration that can control both motor speed and pump displacement. By providing more control variables, the VPVM configuration has opened up new opportunities for improving the performance of EHAs [15]. However, the adoption of VPVM configuration in EHAs will lead to increased system complexity and reduced reliability, which is an unavoidable disadvantage of the VPVM configuration and should be considered when designing EHAs with VPVM configuration (VPVM-EHAs). Jiao et al. [16] proposed a load-sensitive VPVM configuration that directly feeds load pressure back to the variable displacement mechanism. When the load pressure increases, both the pump displacement and motor current decrease, thereby reducing motor heating and improving energy efficiency. However, this configuration will also reduce the stiffness of the EHA when reducing the pump displacement, resulting in a decrease in dynamic response [17]. Yang et al. [18] proposed an active load-sensitive VPVM configuration that adds a proportional pressure reducing valve between the load pressure and the variable displacement mechanism. This configuration can actively control the pump displacement and balance the dynamic response and energy efficiency of the EHA. However, due to the large size and leakage of the proportional pressure reducing valve, this configuration increases the mass and power loss of the EHA [19]. In summary, the dynamic response of the existing VPVM-EHAs is far inferior to that of the traditional FPVM-EHA, which seriously limits the promotion and application of VPVM-EHAs in high-end equipment.

In aviation and aerospace applications, most EHAs in service use symmetrical double-rod cylinders [20]. Compared to dual-rod cylinders, single-rod cylinders offer significant advantages in increasing power density [21] and hold great promise for application in aviation and aerospace fields, which have extremely stringent requirements on EHA power density. However, the unequal effective working area of the rod and rodless chambers in traditional single-rod cylinders leads to the flow mismatch [22]. To solve the flow mismatch problem, some studies have designed asymmetric three-port pumps to match asymmetric single-rod cylinders [23]. However, this solution will increase flow pulsation, noise, and vibration. Some studies have solved the flow mismatch problem by adding a flow compensation valve [24] or an additional pump [25]. However, these solutions will increase the system complexity of EHAs and reduce reliability [26,27,28]. In summary, while existing solutions solve the flow mismatch problem caused by asymmetric single-rod cylinders, they also bring new problems.

This paper designs an innovative VPVM configuration for EHAs by using an electro-hydraulic servo valve (EHSV) for active pump displacement control. This paper also designs an innovative symmetrical single-rod cylinder configuration. By optimizing the internal structure, the effective working areas of the rod and rodless chambers are equal, thus achieving flow matching. Furthermore, this paper develops a high-power density VPVM-EHA prototype employing two innovative configurations. Both simulation and experimental results demonstrate that compared with the traditional FPVM configuration, the proposed VPVM configuration not only significantly improves the energy efficiency of the EHA under low-speed and heavy-load conditions, but also achieves dynamic response performance close to that of the FPVM configuration under rapid response conditions. Furthermore, while maintaining the advantages of high power density, the symmetrical single-rod cylinder perfectly solves the flow mismatch problem of traditional asymmetric single-rod cylinders (without causing new problems), helping the VPVM-EHA prototype achieve a power density of 0.72 kW/kg.

The remainder of this paper is organized as follows. Section 2 introduces the innovative configuration and prototype development of the VPVM-EHA. In Section 3, the mechanism model of VPVM-EHA is established, and simulation analysis of dynamic response is performed. Section 4 conducts experimental comparative tests for the dynamic response and energy efficiency of VPVM-EHA. Conclusions are provided in Section 5.

2. Design and Development

2.1. Configuration Design

As shown in Figure 1, the VPVM-EHA consists of five components, namely a five-phase permanent magnet synchronous motor (F-PMSM), a variable displacement piston pump, an accumulator, a symmetrical single-rod cylinder, and a valve block. When the VPVM-EHA is in operation, the electrical energy is first converted into rotational mechanical energy through the F-PMSM. Then, the F-PMSM drives the piston pump to rotate through the main shaft, converting the rotational mechanical energy into hydraulic energy. Finally, the cylinder converts the hydraulic energy into linear mechanical energy through the pressure difference between two working chambers to drive the piston rod to move [29]. Compared with the traditional FPVM configuration that requires an additional oil source, the VPVM configuration proposed in this paper realizes the variable displacement function by adding an EHSV between the drain port and the cylinder for displacement control. The innovations of the proposed VPVM configuration are: (1) EHSV is introduced into EHA for the first time to realize active displacement control, which is different from traditional fixed displacement or load-sensitive passive displacement control. (2) The external hydraulic source that the existing VPVM configurations rely on is eliminated, and the displacement control is completed based on the pressure of EHA, thereby improving the power density of the EHA and providing a new and efficient displacement control path.

Furthermore, the symmetrical single-rod cylinder configuration perfectly solves the flow mismatch problem caused by traditional asymmetric single-rod cylinder without reducing the power density of VPVM-EHA or causing new problems. As shown in Figure 2, when the F-PMSM rotates forward, port B of the piston pump is connects to port b of the cylinder, driving the piston rod to extend. When the F-PMSM rotates backward, port A of the piston pump connects to port a of the cylinder, driving the piston rod to retract. Under highly dynamic operating conditions, the proposed VPVM configuration can actively increase the displacement to achieve a high dynamic response comparable to that of the traditional FPVM configurations. Under low-speed and heavy-load conditions, the proposed VPVM configuration can actively reduce displacement to achieve a high energy efficiency comparable to that of the existing VPVM configurations. The rotational speed of F-PMSM and the displacement of piston pump jointly control the movement velocity of piston rod.

2.2. Prototype Development

2.2.1. Prototype Parameters

According to the design objectives of the VPVM-EHA prototype, the technical parameters are set, as shown in

Table 1. Technical parameters setting of VPVM-EHA prototype.

Parameters	Values
Max output force	120 kN
Max piston rod velocity (no-load)	110 mm/s
Max stroke	100 mm
Max pressure	28 MPa
Max rotational speed	12,000 rpm
Max torque	14 N·m
Bus voltage	270 VDC
Max flow rate	24 L/min
Max displacement	2 mL/rev

. It is worth noting that the purpose of developing the VPVM-EHA prototype in this paper is solely to validate the effectiveness of the proposed innovative configurations. Rather than using a strict forward-thinking approach, the VPVM-EHA prototype is developed by integrating existing components in the authors’ lab, such as the motor, pump, and valve, while ensuring that all the target functions and performance of the VPVM-EHA are achieved. Therefore, before developing the VPVM-EHA prototype, this paper did not establish detailed calculation models for the technical parameters of key components. Instead, this paper directly referred to calculation models in the existing studies [30] and performed a simple calculation of the technical parameters of key components.

2.2.2. Components Selection

In the aviation and aerospace fields, most in-service EHAs use three-phase motors [31,32]. However, compared with three-phase motors, five-phase motors have excellent fault tolerance and lower output torque ripple [33,34], which has significant advantages in improving the reliability of EHA and has good application prospects in the aviation and aerospace fields where reliability requirements are extremely stringent [35]. This paper selects an oil-cooled F-PMSM prototype with a two-pole and twenty-slot structure, as shown in Figure 3. This F-PMSM prototype is designed and developed by the authors’ laboratory. The F-PMSM prototype with a weight of 6.76 kg, a maximum current of 150 A, a rated voltage of 270 V DC, a maximum current of 150 A, a rated power of 10.22 kW, and a maximum rotational speed of 12,011 rpm. The above technical parameters are actual test results from professional institutions.

This paper selects an axial piston pump prototype with nine pistons. The F-PMSM prototype with a maximum flow of 24 L/min, a maximum pressure of 28 MPa, and a variable displacement range of 0~2 mL/rev. In this paper, an EHSV is added between the cylinder for displacement control and the drain port of piston pump to realize active displacement control. The manufacturer of the selected EHSV is Zhejiang Hanqin Technology Co., Ltd., (Hangzhou, China) and the model is CSDX3-2A. For the selected EHSV, the weight is 176 g, the maximum operating pressure is 40 MPa, the response time is 40 ms (when the valve receives an input signal, it takes about 40 ms to complete the action and stabilize the output), the output pressure range is 0~40 MPa, and the maximum control current is 40 mA. Since the actuation frequency of the VPVM-EHA prototype is much lower than the bandwidth of the EHSV and the response time of the EHSV is fast enough (only 40 ms), the EHSV has little impact on the system dynamic response of the VPVM-EHA and can meet the dynamic response requirements of the system.

As shown in Figure 4, by adjusting the control current of the EHSV, the pressure difference between the two working chambers of the cylinder for displacement control can be adjusted, thereby adjusting the deflection angle of the swash plate by extending or retracting the piston rod, and finally achieving the goal of adjusting the displacement of the piston pump. The damper connected to the spring provides restoring force and stable support for the swash plate, ensuring that the swash plate can return to the default angle when the hydraulic control force disappears or reaches equilibrium, thereby guaranteeing the safety and stability of the piston pump. To improve the reliability of the variable displacement mechanism as much as possible, the swash plate angle sensor is not installed on the piston pump. This paper directly sets the displacement of the piston pump to be positively correlated with the control current of the EHSV, that is, the larger the control current, the larger the pump displacement [36].

In addition, to further improve the reliability of the VPVM-EHA prototype, this paper eliminats the traditional shaft-end seal between the F-PMSM and the piston pump through optimized design, which not only reduced friction and power loss but also simplified the system structure [37].

Based on the symmetrical single-rod cylinder configuration proposed in Section 2.1, this paper further developed a symmetrical single-rod cylinder prototype, as shown in Figure 5. This cylinder adopts a coaxial dual-chamber structure with equal effective working areas (Area I = Area II in Figure 5), ensuring symmetrical oil flow during piston rod extension and retraction [38]. A displacement sensor (the manufacturer is Qijian and the model is AC-50-AS) is installed within the cylinder to monitor the position of piston rod in real time. The cylinder has three oil ports, namely port a, port b, and port T. When oil enters port a, the piston rod extends. When oil enters port b, the piston rod retracts. Port T is connected to the air chamber to compensate for the volume difference between the inlet and outlet oil, thus preventing pressure fluctuations within the cylinder.

By integrating the above selected/designed F-PMSM prototype, axial piston pump prototype with variable displacement function, and symmetrical single-rod cylinder prototype, as well as the high-pressure accumulator and valve block, this paper develops a VPVM-EHA prototype, as shown in Figure 6. The VPVM-EHA prototype with a power density of 0.72 kW/kg.

3. Dynamic Modeling and Simulation

3.1. Dynamic Model

To preliminarily analyze the improvement in dynamic performance achieved by the VPVM configuration and design a controller to conduct functional and performance experiments of the VPVM-EHA prototype, a simple dynamic model of VPVM-EHA is established.

3.1.1. F-PMSM Model

The voltage model of F-PMSM is established as [39]

$$U_m=R_m{i}_m+L_{m}d{i}_m/dt+K_e{\omega }_m,$$

(1)

where $U_m$ and $i_m$ are the armature voltage and current, respectively. $R_m$ is the resistance, $L_m$ is the inductance, ${\omega }_m$ is the angular speed, $K_e={\displaystyle \frac{\rho N{\omega }_m\phi }{60a}}$ is the back electromotive force (EMF) coefficient, $\rho$ is the number of pole pairs, $N$ is the total number of conductors, $\phi$ is the magnetic flux, and $a$ is the number of branch pairs.

The torque model of F-PMSM is established under the assumption that cogging torque, iron losses, and mechanical friction are neglected. The electromagnetic torque can be calculated as [40]

$$T_m=J_m{\displaystyle \frac{d{\omega }_m}{dt}}+B_m{\omega }_m+T_L=K_T{i}_m,$$

(2)

where $T_m$ is the electromagnetic torque, $B_m$ and $J_m$ are the damping factor and moment of the motor-pump group inertia, respectively. $B_m$ and $J_m$ are mainly determined by the mass of permanent magnet rotors in F-PMSM, as well as the mass and number of pistons in the piston pump. $T_L$ is the load torque and $K_T$ is the torque coefficient.

The relationship between ${\omega }_m$, $U_m$ and $T_L$ in the Laplace domain can be obtained by combining Equations (1) and (2), as follows

$$\begin{array}{l}{\omega }_m\left(s\right)={\displaystyle \frac{K_T{U}_m(s)-(R_m+L_{m}s)T_L}{L_m{J}_m{s}^2+(R_m{J}_m+L_m{B}_m)s+(R_m{B}_m+K_e{K}_T)}}\\ ={\displaystyle \frac{\frac{K_T}{R_m{B}_m+K_e{K}_T}{U}_m(s)-\frac{R_m}{R_m{B}_m+K_e{K}_T}(1+\frac{L_m}{R_m})T_L}{\frac{s^2}{{\varpi }_m^2}+\frac{2{\xi }_m}{{\varpi }_m}s+1}}\end{array},$$

(3)

where ${\varpi }_m=\sqrt{{\displaystyle \frac{R_m{B}_m+K_e{K}_T}{L_m{J}_m}}}$ is the natural frequency of F-PMSM and ${\xi }_m={\displaystyle \frac{R_m{J}_m+L_m{B}_m}{2\left(R_m{B}_m+K_e{K}_T\right)}}\sqrt{{\displaystyle \frac{R_m{B}_m+K_e{K}_T}{L_m{J}_m}}}$ is the relative damping ratio of F-PMSM.

3.1.2. Piston Pump Model

The flow continuity equation of piston pump is established as [41]

$$\begin{array}{l}{Q}_{\mathrm{A}}=D_p{\omega }_m-C_{ipp}(p_{\mathrm{A}}-p_{\mathrm{B}})-C_{epp}{p}_{\mathrm{A}}-{\displaystyle \frac{V_{\mathrm{A}}}{{\beta }_e}}{\displaystyle \frac{d{p}_{\mathrm{A}}}{dt}}\\ Q_{\mathrm{B}}=D_p{\omega }_m-C_{ipp}(p_{\mathrm{A}}-p_{\mathrm{B}})+C_{epp}{p}_{\mathrm{B}}+{\displaystyle \frac{V_{\mathrm{B}}}{{\beta }_e}}{\displaystyle \frac{d{p}_{\mathrm{B}}}{dt}}\end{array},$$

(4)

where $Q_{\mathrm{A}}$ and $Q_{\mathrm{B}}$ are the flow rates of ports A and B, respectively. $D_p$ is the pump displacement, $p_A$ and $p_B$ are the pressures of ports A and B, respectively. $V_{\mathrm{A}}$ and $V_{\mathrm{B}}$ are the initial volumes of chambers connected to ports A and B, respectively. $C_{ipp}$ and $C_{epp}$ are the internal leakage coefficient and the external leakage coefficient, respectively. ${\beta }_e$ is the effective bulk elastic modulus.

The flow equation of variable displacement mechanism can be established as

$$q_v=C_d\varpi x_v\sqrt{{\displaystyle \frac{2}{\rho }}}\mathrm{sgn}(\Delta p)\sqrt{\left|\Delta p\right|},$$

(5)

where $q_v$ is the flow rate of EHSV, $C_d$ is the discharge coefficient, $\varpi$ is the orifice width, $x_v$ is the valve core displacement, $\rho$ is the oil density, $\Delta p$ is the orifice pressure drop.

When $x_v$ is small and the pressure difference is approximately constant, Equation (5) can be linearized as

$$q_v=K_q{x}_v-K_c{p}_L=A_f{\dot{x}}_f+C_{ipf}{p}_v+C_{epf}{p}_v+{\displaystyle \frac{V_f}{{\beta }_e}}{\displaystyle \frac{d{p}_v}{dt}},$$

(6)

where $K_q={\displaystyle \frac{\mathsf{\partial }{q}_v}{\mathsf{\partial }{x}_v}}$ is the flow gain, $K_c={\displaystyle \frac{\mathsf{\partial }{q}_v}{\mathsf{\partial }{p}_v}}$ is the flow-pressure coefficient. In the cylinder for displacement control, $A_f$ is the effective area of working chamber, ${\dot{x}}_f$ is the piston rod velocity, $C_{ipf}$ and $C_{epf}$ are the internal and external leakage coefficients, respectively. $p_v$ is the EHSV output pressure and $V_f$ is the initial volume of working chamber.

Balance equations of the cylinder for displacement control and load forces can be established under the assumption of pure viscous friction, as follows

$$A_f{p}_v=m_f{\ddot{x}}_f+B_f{\dot{x}}_f+F_f,$$

(7)

where $m_f$ is the load mass, ${\ddot{x}}_f$ is the rod acceleration, $B_f$ is the friction coefficient, and $F_f$ is the external load force.

Since the initial displacement of the piston pump is the maximum displacement, the relationship between $D_p$, $i_f$ and $F_f$ in the Laplace domain can be obtained by combining Equations (4)–(7), as follows

$$D_p\left(s\right)=D_{\mathit{pmax}}-{\displaystyle \frac{\frac{K_c{K}_{q}K{i}_f\left(s\right)}{A_f}-\frac{K_c}{A_f^2}\left(1+\frac{V_f}{4{\beta }_e}s\right)F_f}{s(\frac{s^2}{{\varpi }_h^2}+\frac{2{\xi }_h}{{\varpi }_h}s+1)}},$$

(8)

where $x_v=K{i}_f$, $i_f$ is the control current of EHSV, $K$ is the current gain, and $D_{\mathit{pmax}}$ is the maximum displacement of piston pump. ${\varpi }_h=\sqrt{{\displaystyle \frac{4{\beta }_e{A}_f^2}{V_f{m}_f}}}$ is the natural frequency of the variable displacement mechanism and ${\xi }_h={\displaystyle \frac{K_c+C_{ipf}+\frac{1}{2}{C}_{epf}}{A_f}}\sqrt{{\displaystyle \frac{{\beta }_e{m}_f}{V_f}}}+{\displaystyle \frac{B_f}{4A_f}}\sqrt{{\displaystyle \frac{V_f}{{\beta }_e{m}_f}}}$ is the relative damping ratio of the variable displacement mechanism.

3.1.3. Cylinder Model

Based on the assumption of laminar flow, the nonlinear orifice equation can be linearized [42]. When the piston rod extends, the flow rates of the inlet chamber $q_1$ and the outlet chamber $q_2$ are

$$\begin{array}{l}{q}_1=A{\dot{x}}_p+C_{ipc}(p_1-p_2)+C_{epc}{p}_1+{\displaystyle \frac{V}{{\beta }_e}}{\displaystyle \frac{d{p}_1}{dt}}\\ q_2=A{\dot{x}}_p+C_{ipc}(p_1-p_2)-C_{epc}{p}_2-{\displaystyle \frac{V}{{\beta }_e}}{\displaystyle \frac{d{p}_2}{dt}}\end{array},$$

(9)

where ${\dot{x}}_p$ is the rod velocity, $C_{ipc}$ and $C_{epc}$ are the internal leakage coefficient and external leakage coefficient, respectively. $p_1$ and $p_2$ are the pressure of two working chambers, $V$ is the initial volume of the working chamber, and $A$ is the effective area of the working chamber.

When the piston rod retracts, the flow rates of the inlet chamber $q_2$ and the outlet chamber $q_1$ are

$$\begin{array}{l}{q}_1=A{\dot{x}}_p+C_{ipc}(p_2-p_1)-C_{epc}{p}_1-{\displaystyle \frac{V}{{\beta }_e}}{\displaystyle \frac{d{p}_1}{dt}}\\ q_2=A{\dot{x}}_p+C_{ipc}(p_2-p_1)+C_{epc}{p}_2+{\displaystyle \frac{V}{{\beta }_e}}{\displaystyle \frac{d{p}_2}{dt}}\end{array},$$

(10)

The force balance equation of the symmetrical single-rod cylinder can be established under the assumption of pure viscous friction, as follows

$$p_L={\displaystyle \frac{F_{out}}{A}}={\displaystyle \frac{M_c{\ddot{x}}_p+B_c{\dot{x}}_p+F_{ex}}{A}},$$

(11)

where $F_{out}$ is the output force, $M_c$ is the external load mass, ${\ddot{x}}_p$ is the rod acceleration, $B_c$ is the friction coefficient, and $F_{ex}$ is the external load.

By combining Equations (4), (9) and (10), ignoring external leakage and pump oil volume deformation, the relationship between $X_p$ and ${\omega }_m$ in the Laplace domain is obtained as

$$\begin{array}{l}{\displaystyle \frac{X_p(s)}{{\omega }_m(s)}}={\displaystyle \frac{2D_p}{\frac{V{M}_c}{{\beta }_e\mathrm{A}}{s}^3+\frac{2M_c{\beta }_e\left(C_{ipp}+C_{ipc}\right)+V{B}_c}{{\beta }_e\mathrm{A}}{s}^2+\left[2A+\frac{2B_c\left(C_{ipp}+C_{ipc}\right)}{\mathrm{A}}\right]s}}\\ ={\displaystyle \frac{\frac{2\mathrm{A}{D}_p}{\left[{\mathrm{A}}^2+\left(C_{ipp}+C_{ipc}\right)B_c\right]}}{s(\frac{s^2}{{\varpi }_{hc}^2}+\frac{2{\xi }_{hc}}{{\varpi }_{hc}}s+1)}}\end{array},$$

(12)

The natural frequency of the symmetrical single-rod cylinder is

$${\varpi }_{hc}=\sqrt{{\displaystyle \frac{2{\beta }_e\left[{\mathrm{A}}^2+\left(C_{ipp}+C_{ipc}\right)B_c\right]}{\mathrm{V}{M}_c}}},$$

(13)

The relative damping ratio is

$${\xi }_{hc}={\displaystyle \frac{\mathrm{V}{B}_c+2{\beta }_e\left(C_{ipp}+C_{ipc}\right)M_c}{4{\beta }_e\left[{\mathrm{A}}^2+\left(C_{ipp}+C_{ipc}\right)B_c\right]}}\sqrt{{\displaystyle \frac{2{\beta }_e\left[{\mathrm{A}}^2+\left(C_{ipp}+C_{ipc}\right)B_c\right]}{\mathrm{V}{M}_c}}},$$

(14)

3.1.4. VPVM-EHA Model

Based on the dynamic models of key components established above, the relationship between $X_p$, $U_m$, and $i_f$ in the Laplace domain can be established as

$$\begin{array}{l}{\displaystyle \frac{X_p(s)}{U_m(s)i_f\left(s\right)}}={\displaystyle \frac{{\omega }_m\left(s\right)}{U_m(s)}}\times {\displaystyle \frac{D_p\left(s\right)}{i_f\left(s\right)}}\times {\displaystyle \frac{X_p(s)}{{\omega }_m\left(s\right)D_p\left(s\right)}}\\ ={\displaystyle \frac{\frac{K_T}{R_m{B}_m+K_e{K}_T}}{\frac{s^2}{{\varpi }_m^2}+\frac{2{\xi }_m}{{\varpi }_m}s+1}}\times \left(D_{p\max }-{\displaystyle \frac{\frac{{\overline{K}}_p{K}_{q}K{i}_f\left(s\right)}{A_f}}{s(\frac{s^2}{{\varpi }_h^2}+\frac{2{\xi }_h}{{\varpi }_h}s+1)}}\right)\times {\displaystyle \frac{\frac{2\mathrm{A}{D}_p}{\left[{\mathrm{A}}^2+\left(C_{ipp}+C_{ipc}\right)B_c\right]}}{s(\frac{s^2}{{\varpi }_{hc}^2}+\frac{2{\xi }_{hc}}{{\varpi }_{hc}}s+1)}}\end{array}$$

(15)

The block diagram of the EHA system in the Laplace domain is shown in Figure 7, and the model parameter settings are shown in

Table 2. Parameter settings for the dynamic model of VPVM-EHA.

Components	Parameters	Values
F-PMSM	$R_m(\mathrm{\Omega })$	0.0375
	$L_m(H)$	0.022
	$J(kg\cdot m^2)$	4.8 × 10−4
	${\psi }_f(Wb)$	0.01235
	$\rho$	4
	$B_m(N. m. s. ra{d}^{-1})$	0.001
	$U_m(V)$	270
Piston pump	$D_{p\max }(mL/rev)$	2
	$C_{ipp}(m^3. s^{-1}. p{a}^{-1})$	1 × 10−13
Variable displacement mechanism	$m_t(kg)$	0.1
	$K_s(N/m)$	20,000
	$A_{\mathrm{f}}(m^2)$	1.76 × 10−7
	$C_{ipf}(m^3. s^{-1}. p{a}^{-1})$	1 × 10−13
	$x_{v\max }(mm)$	4
	$B_f(N/(m/s))$	100
Cylinder	$C_{ipc}(m^3. s^{-1}. p{a}^{-1})$	1 × 10−13
	$A(m^2)$	4615 × 10−6
	$x_{p\max }(mm)$	100
	${\beta }_e(MPa)$	1000
	$M(kg)$	400
	$B_c(N/(m/s))$	200

. Furthermore, the Bode diagram of VPVM-EHA can be plotted with the highest frequency of 10.75 Hz, as shown in Figure 8.

3.2. Controller Design

Figure 9 shows the block diagram of the VPVM-EHA controller. The VPVM-EHA is a dual-input and single-output system. The controller consists of three closed-loop control loops, namely the positional control loop, speed control loop and the displacement control loop. For the displacement control loop, under highly dynamic operating conditions, the controller actively increases the displacement to improve the dynamic response of VPVM-EHA. Under low-speed and heavy-load conditions, the controller actively reduces the displacement to reduce motor current, thereby improving the energy efficiency of VPVM-EHA. For the speed control loop, the controller dynamically adjusts the rotational speed of F-PMSM based on the load demand, thereby matching the flow rate and output power requirements. Both control loops utilize optimized PID controllers, with parameter settings shown in

Table 3. Parameter settings of controller.

Controllers	Parameters	Values
Positional controller	${\mathrm{K}}_{p1}$	1 × 107
	${\mathrm{K}}_{I1}$	1 × 104
	${\mathrm{K}}_{D1}$	0
Speed controller	${\mathrm{K}}_{p2}$	0.2
	${\mathrm{K}}_{I2}$	10
	${\mathrm{K}}_{D2}$	0
Displacement controller	${\mathrm{K}}_{p3}$	1 × 103
	${\mathrm{K}}_{I3}$	50
	${\mathrm{K}}_{D3}$	0

. It should be noted that the dynamic model established in this paper is solely for the purpose of developing a simple PID control method for functional and performance experiments on the VPVM-EHA prototype. Therefore, to reduce the complexity of modeling and controller design, some nonlinear models are significantly simplified and linearized, and nonlinear factors such as the oil elastic modulus, friction, leakage, and moment of inertia are ignored.

3.3. Dynamic Performance

In this paper, a step displacement condition with a displacement amplitude of 10 mm and a constant load of 40 kN is designed to analyze the dynamic response performance of the proposed VPVM configuration compared to the traditional FPVM configuration. The FPVM configuration is implemented by setting the displacement of the VPVM-EHA to a constant value of 2 mL/rev.

Simulation results for this condition are shown in Figure 10. The step rise time of the FPVM configuration is 0.08 s, while that of the VPVM configuration is 0.09 s. The VPVM configuration maintains a low pump displacement before the step displacement command (at 0.3 s) to achieve higher energy efficiency. Once the step displacement occurs, the pump displacement increases rapidly, thereby improving the dynamic response. The difference of 0.01 s in rise time between the VPVM and FPVM configurations corresponds to the response delay of the cylinder for displacement control. The changes in pump displacement is shown in Figure 11.

In addition, the motor speed and motor current of the VPVM configuration are shown in Figure 12 and Figure 13. It can be observed that the motor speed rises sharply at 0.3 s, reaching nearly 12,000 rpm, which is the maximum allowable speed and thus approaching saturation. Similarly, the motor current increases similar to the motor speed, reaching the maximum allowable value of 150 A, also approaching saturation.

The above simulation results show that under the step displacement condition, the VPVM configuration not only achieves dynamic response performance comparable to that of the FPVM configuration, but can also operates close to the physical saturation boundaries of the F-PMSM.

4. Results and Discussions

4.1. Experiment Setup

The EHA experimental platform consists of the EHA system, the real-time simulation system, and the drive system, as shown in Figure 14. The EHA system consists of the VPVM-EHA and the loading system. The loading system uses a valve-controlled cylinder to apply external loading force to the VPVM-EHA.

The symmetrical single-rod cylinder uses a built-in LVDT displacement sensor from Qijian, Nanjing, China (model AC-50-AS). The angle sensor of F-PMSM is from Tamagawa, Shanghai, China (model TS2620N21E11), and the current sensor of F-PMSM is from LEM, Shanghai, China (model CASR-50). All sensor data is pre-processed using a low-pass filter with a passband frequency range of 0.002~0.005 Hz. The EHSV is powered by a 24 V DC power supply (the manufacturer is IVYTECH, Dongwan, China and the model is IV3033D).

The real-time simulation system uses Dspace to run the MATLAB 2022 program, collects the cylinder position, motor speed, and motor current signals as inputs (the sampling time is 0.125 ms), and outputs the motor speed and pump displacement control signals. The drive system uses an IGBT motor driver with an input voltage of 270 V DC.

The experimental platform operates as follows: First, the MATLAB control program on the host computer is compiled to the real-time simulation system. The real-time simulation system then outputs the motor speed and pump displacement control signals to the drive system. After that, the drive system outputs the control currents of F-PMSM and EHSV, and the VPVM-EHA begins operation. During the operation of VPVM-EHA, the real-time simulation system collects the cylinder position and motor current signals and inputs them into the MATLAB control program. The MATLAB control program outputs the motor speed and pump displacement control signals, which are then transmits to the drive system. The drive system outputs the control currents of F-PMSM and EHSV. Both F-PMSM and EHSV use PID controllers.

4.2. Experiment Design

To better highlight the performance of the proposed VPVM configuration, a fair comparison is conducted with the traditional FPVM configuration. The pump displacement of FPVM configuration is set to the maximum displacement of 2 mL/rev.

A sinusoidal displacement condition with a displacement amplitude of 45 mm, a sinusoidal frequency of 0.1 Hz, and a constant load of 100 kN is designed. This condition is a typical low-speed and heavy-load condition, and is utilized to analyze the energy efficiency improvement of the proposed VPVM configuration compared with the traditional FPVM configuration. The step displacement condition utilized to verify the dynamic response is exactly the same as the simulation condition designed in Section 3.3. This condition is utilized to verify the dynamic response performance of the proposed VPVM configuration compared to that of the traditional FPVM configuration. The sinusoidal displacement condition is referred to as C1 and the step displacement condition is referred to as C2.

4.3. Experimental Results

In this paper, the trajectory tracking error and the maximum absolute percentage error (MAPE) is also used to characterize the control performance.

Figure 15 shows the comparative experimental results under C1. Figure 15a,b show that both VPVM-EHA and FPVM-EHA track the reference trajectory well, with the VPVM-EHA achieving higher position control accuracy. As shown in Figure 15d, the MAPEs of VPVM-EHA and FPVM-EHA are 0.61% and 0.87%, respectively. Figure 15c shows the motor heating results under C1. The copper loss heat generation of VPVM-EHA and FPVM-EHA are 4.23× 10⁴ J and 9.19× 10⁴ J, respectively. The heat generated by VPVM-EHA is only 46.03% of that of FPVM-EHA.

Figure 16 shows the changes in pump displacement of VPVM-EHA and FPVM-EHA under C1. It can be found that the VPVM-EHA can reduce motor heat generation and improve overall energy efficiency by actively reducing pump displacement.

Figure 17 and Figure 18 show the experimental results under C2. As shown in Figure 17a, the step rise times of the VPVM-EHA and FPVM-EHA are 0.09 s and 0.08 s, respectively. highly consistent with the simulation results. The experimental results in Figure 17a are highly consistent with the simulation results in Figure 10, verifying the accuracy of the VPVM-EHA model. In Figure 17c, the copper loss heat generation of the VPVM-EHA and FPVM-EHA are 790 J and 930 J, respectively, and the heat generation of the VPVM-EHA is only 84.9% of that of the FPVM-EHA. In Figure 17d, the MAPEs of the VPVM-EHA and FPVM-EHA are 1.1% and 3.4%, respectively, further highlighting the improved control performance of the VPVM-EHA. The results in Figure 17 demonstrate that the VPVM-EHA effectively reduces motor heat generation while maintaining a dynamic response similar to that of the FPVM-EHA.

Figure 18 shows the changes in pump displacement of VPVM-EHA and FPVM-EHA under C2. It can be found that the VPVM-EHA actively increases pump displacement during the step process to improve dynamic response, and reduces pump displacement after the step to improve energy efficiency.

Furthermore, the experimental results fully demonstrate that the proposed symmetric single-rod cylinder configuration can perfectly solve the flow mismatch problem caused by traditional asymmetric single-rod cylinders without reducing the power density of the VPVM-EHA or causing new problems.

5. Conclusions

To meet the demand for improving the power density, energy efficiency, and dynamic response of EHAs, this paper first proposes an innovative VPVM configuration and a symmetrical single-rod cylinder configuration. Then, a corresponding VPVM-EHA prototype is developed, achieving a power density of up to 0.72 kW/kg. Next, a simple dynamic model is established to conduct preliminary simulation analysis for the dynamic response of the VPVM configuration and design a controller for the VPVM-EHA. The results demonstrate that the VPVM configuration achieves dynamic response performance very close to that of the FPVM configuration and higher control accuracy, while significantly improving energy efficiency by actively controlling the pump displacement. The dynamic response simulation results are highly consistent with the experimental results, verifying the accuracy of the dynamic model. Furthermore, the experimental results fully demonstrate that the proposed symmetric single-rod cylinder configuration can perfectly solve the flow mismatch problem caused by traditional asymmetric single-rod cylinders without reducing the power density of the VPVM-EHA or causing new problems. The results of this paper can be utilized to improve the power density, dynamic response, and energy efficiency of EHAs. In future work, based on the VPVM-EHA prototype, the authors will consider various nonlinear factors to develop a more refined model and design a controller with improved performance by considering various uncertain parameters and disturbances.